Dear @Jassica,
Welcome back to the Wefugees Community.
I totally understand that your situation is quite frustrating - I am sorry to hear.
From my experience, however, the process of moving to another city during the asylum procedure is really complicated and only granted in exceptional cases (the approval of a so-called "Umverteilungsantrag"). I am afraid, that a minijob offer might not convince the authorities to give you the permission. To assess your individual case as a whole, I recommend you to contact a counselling office (as you already thought of). This search engine can help to find respective services in your area. If you want to share your approximate location, I am also happy to make some research.
In any case, please don't hesitate to get back to us with any further questions.
All the best,
Meike